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We identify the motivicKGL/2-local sphere as the fiber of\psi^{3}-1on(2,\eta)-completed HermitianK-theory, over any base scheme containing1/2. This is a motivic analogue of the classical resolution of theK(1)-local sphere, and extends to a description of theKGL/2-localization of an arbitrary motivic spectrum. Our proof relies on a novel conservativity argument that should be of broad utility in stable motivic homotopy theory.
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The slice spectral sequence for a motivic analogue of the connective K(1)$K(1)$‐local sphere
Abstract We compute the 2‐adic effective slice spectral sequence (ESSS) for the motivic stable homotopy groups of , a motivic analogue of the connective ‐local sphere over prime fields of characteristic not two. Together with the analogous computation over algebraically closed fields, this yields information about the motivic ‐local sphere over arbitrary base fields of characteristic not two. To compute the spectral sequence, we prove several results that may be of independent interest. We describe the ‐differentials in the slice spectral sequence in terms of the motivic Steenrod operations over general base fields, building on analogous results of Ananyevskiy, Röndigs, and Østvær for the very effective cover of Hermitian K‐theory. We also explicitly describe the coefficients of certain motivic Eilenberg–MacLane spectra and compute the ESSS for the very effective cover of Hermitian K‐theory over prime fields.
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- PAR ID:
- 10556460
- Publisher / Repository:
- London Mathematical Society
- Date Published:
- Journal Name:
- Proceedings of the London Mathematical Society
- Volume:
- 129
- Issue:
- 5
- ISSN:
- 0024-6115
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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